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Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
* [[Compton]] (+225/224) → [[Compton family #Compton|Compton family]]
* [[Misty]] (+3136/3125 or 5120/5103) → [[Misty family]]
* ''[[Term (temperament)|Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Sextile]]'' (+33554432/33480783) → [[Garischismic clan #Sextile|Garischismic clan]]
* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]
* ''[[Terture]]'' (+359661568/358722675) → [[Vulture family #Terture|Vulture family]]
* ''[[Pentadecal]]'' (+875/864) → [[Keemic temperaments #Pentadecal|Keemic temperaments]]
* ''[[Chromat]]'' (+10976/10935) → [[Amity family #Chromat|Amity family]]
* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Tritikleismic]]'' (+1029/1024) → [[Kleismic family #Tritikleismic|Kleismic family]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Caleb]]'' (+33075/32768) → [[Mirwomo temperaments #Caleb|Mirwomo temperaments]]
* ''[[Trisensory]]'' (+1728/1715) → [[Sensipent family #Trisensory|Sensipent family]]
* ''[[Trisensory]]'' (+1728/1715) → [[Sensipent family #Trisensory|Sensipent family]]
* [[Ennealimmal]] (+2401/2400 or 4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* [[Misty]] (+3136/3125 or 5120/5103) → [[Misty family]]
* ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]
* ''[[Nessafof]]'' (+6144/6125) → [[Porwell temperaments #Nessafof|Porwell temperaments]]
* ''[[Chromat]]'' (10976/10935) → [[Hemimage temperaments #Chromat|Hemimage temperaments]]
* ''[[Term (temperament)|Term]]'' (+32805/32768) → [[Schismatic family #Term|Schismatic family]]
* ''[[Caleb]]'' (+33075/32768) → [[Mirwomo temperaments #Caleb|Mirwomo temperaments]]
* [[Mutt]] (+65625/65536) → [[Horwell temperaments #Mutt|Horwell temperaments]]
* [[Mutt]] (+65625/65536) → [[Horwell temperaments #Mutt|Horwell temperaments]]
* ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic family]]
* ''[[Triquart]]'' (+117649/116640) → [[Quartonic family #Triquart|Quartonic family]]
* ''[[Stearnscape]]'' (+118098/117649) → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Stearnscape]]'' (+118098/117649) → [[Stearnsmic clan #Stearnscape|Stearnsmic clan]]
* ''[[Domain (temperament)|Domain]]'' (645700815/645657712) → [[Minortonic family #Domain|Minortonic family]]
* ''[[Trivish]]'' (+134217728/133984375) → [[Vishnu family #Trivish|Vishnu family]]
* ''[[Tritricot]]'' (+{{monzo| 35 -23 -3 3 }}) → [[Alphatricot family #Tritricot|Alphatricot family]]
* ''[[Domain (temperament)|Domain]]'' (+645700815/645657712) → [[Minortone family #Domain|Minortone family]]
* ''[[Aemilic]]'' (+{{monzo|-84 53}}) → [[159th-octave temperaments #Aemilic|159th-octave temperaments]]
* [[Compton]] (+225/224) → [[Compton family #Compton|Compton family]]
* ''[[Aemilic]]'' (+{{monzo|-84 53}}) → [[Mercator family #Aemilic|Mercator family]]


Considered below are sextile, septichrome, pnict, atomic, avicenna, terture, slendscape, akjayland, magnesium, chromium, zinc, and poe.
Considered below are septichrome, akjayland, pnict, atomic, slendscape, avicenna, zinc, magnesium, poe, chromium, and phosphorus in the order of increasing [[badness]].
 
== Sextile ==
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Sextile (5-limit)]].''
 
Sextile tempers out the [[garischisma]] with a 1/6-octave period and is the {{nowrap| 12 & 270 }} temperament.  


== Septichrome ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 33554432/33480783
[[Comma list]]: 250047/250000, 2460375/2458624


{{Mapping|legend=1| 6 0 71 150 | 0 1 -6 -14 }}
{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
: mapping generators: ~4096/3645, ~3
: mapping generators: ~63/50, ~243/224


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~4096/3645 = 199.9828{{c}}, ~3/2 = 702.1521{{c}}
* [[WE]]: ~63/50 = 400.0100{{c}}, ~243/224 = 140.3702{{c}}
: [[error map]]: {{val| -0.103 +0.094 +0.173 -0.088 }}
: [[error map]]: {{val| +0.030 -0.074 -0.010 +0.059 }}
* [[CWE]]: ~4096/3645 = 200.0000{{c}}, ~3/2 = 702.2187{{c}}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~243/224 = 140.3685{{c}}
: error map: {{val| 0.000 +0.264 +0.374 +0.112 }}
: error map: {{val| 0.000 -0.113 -0.050 +0.017 }}


{{Optimal ET sequence|legend=1| 12, …, 258, 270, 1362c, 1632c, …, 2442bc, 2712bc }}
{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}


[[Badness]] (Sintel): 1.77
[[Badness]] (Sintel): 0.426


=== 11-limit ===
=== Semiseptichrome ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 5632/5625, 9801/9800, 151263/151250
Comma list: 9801/9800, 151263/151250, 234375/234256


Mapping: {{mapping| 6 0 71 150 230 | 0 1 -6 -14 -22 }}
{{Mapping|legend=0| 6 1 -15 -24 -32 | 0 5 17 24 31 }}
: mapping generators: ~55/49, ~375/308


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 199.9817{{c}}, ~3/2 = 702.1383{{c}}
* WE: ~55/49 = 200.0058{{c}}, ~375/308 = 340.3742{{c}} (~1760/1701 = 59.6375{{c}})
* POTE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2080{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3661{{c}} (~1760/1701 = 59.6339{{c}})


{{Optimal ET sequence|legend=0| 12, …, 258e, 270, 822, 1092, 1362c }}
{{Optimal ET sequence|legend=0| 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc }}


Badness (Sintel): 0.981
Badness (Sintel): 0.642


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 5632/5625, 10648/10647
Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184


Mapping: {{mapping| 6 0 71 150 230 279 | 0 1 -6 -14 -22 -27 }}
{{Mapping|legend=0| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9804{{c}}, ~3/2 = 702.1260{{c}}
* WE: ~55/49 = 199.9936{{c}}, ~375/308 = 340.3707{{c}} (~121/117 = 59.6165{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2001{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3802{{c}} (~121/117 = 59.6198{{c}})


{{Optimal ET sequence|legend=0| 12f, …, 258ef, 270, 552, 822, 1092, 1914cde }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Sintel): 0.788
Badness (Sintel): 1.64


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 936/935, 1701/1700, 1716/1715, 5632/5625, 7744/7735
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952


Mapping: {{mapping| 6 0 71 150 230 279 -4 | 0 1 -6 -14 -22 -27 3 }}
{{Mapping|legend=0| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9669{{c}}, ~3/2 = 702.0643{{c}}
* WE: ~55/49 = 199.9865{{c}}, ~375/308 = 340.3619{{c}} (~88/85 = 59.6111{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1869{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3821{{c}} (~88/85 = 59.6179{{c}})


{{Optimal ET sequence|legend=0| 12f, 270, 552g }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Sintel): 1.06
Badness (Sintel): 1.39


==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 936/935, 1216/1215, 1701/1700, 1716/1715, 2376/2375, 4200/4199
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408


Mapping: {{mapping| 6 0 71 150 230 279 -4 35 | 0 1 -6 -14 -22 -27 3 -1 }}
{{Mapping|legend=0| 6 1 -15 -24 -32 -68 -1 34 | 0 5 17 24 31 53 15 -5 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/49 = 199.9711{{c}}, ~3/2 = 702.0829{{c}}
* WE: ~55/49 = 199.9837{{c}}, ~162/133 = 340.3589{{c}} (~88/85 = 59.6084{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.1890{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})


{{Optimal ET sequence|legend=0| 12f, 270, 552g, 822gg }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Sintel): 0.948
Badness (Sintel): 1.35


=== Sextilia ===
==== 23-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 1001/1000, 4096/4095, 4459/4455, 20449/20412
Comma list: 936/935, 1701/1700, 1716/1715, 1863/1862, 2024/2023, 2025/2023, 2376/2375


Mapping: {{mapping| 6 0 71 150 230 -149 | 0 1 -6 -14 -22 18 }}
{{Mapping|legend=0| 6 1 -15 -24 -32 -68 -1 34 -12 | 0 5 17 24 31 53 15 -5 23 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9975{{c}}, ~3/2 = 702.2196{{c}}
* WE: ~55/49 = 199.9829{{c}}, ~162/133 = 340.3576{{c}} (~88/85 = 59.6081{{c}})
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2285{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})


{{Optimal ET sequence|legend=0| 12, 258e, 270 }}
{{Optimal ET sequence|legend=0| 282, 342f, 624 }}


Badness (Sintel): 1.62
Badness (Sintel): 1.14


==== 17-limit ====
== Akjayland ==
Subgroup: 2.3.5.7.11.13.17
{{See also| 21st-octave temperaments }}


Comma list: 715/714, 1001/1000, 1701/1700, 4096/4095, 4459/4455
Named by [[Eliora]] in 2022, akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.  
 
Mapping: {{mapping| 6 0 71 150 230 -149 -4 | 0 1 -6 -14 -22 18 3 }}
 
Optimal tunings:
* WE: ~55/49 = 199.9862{{c}}, ~3/2 = 702.1714{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2207{{c}}
 
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}
 
Badness (Sintel): 1.95


==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 715/714, 1001/1000, 1216/1215, 1701/1700, 1729/1728, 2912/2907
Mapping: {{mapping| 6 0 71 150 230 -149 -4 35 | 0 1 -6 -14 -22 18 3 -1 }}
Optimal tunings:
* WE: ~55/49 = 199.9866{{c}}, ~3/2 = 702.1731{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 702.2208{{c}}
{{Optimal ET sequence|legend=0| 12, 258e, 270 }}
Badness (Sintel): 1.53
== Septichrome ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 2460375/2458624
[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}


{{Mapping|legend=1| 3 3 1 0 | 0 5 17 24 }}
{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
: mapping generators: ~63/50, ~243/224
: mapping generators: ~1323/1280, ~131072/91875


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 400.0100{{c}}, ~243/224 = 140.3702{{c}}
* [[WE]]: ~1323/1280 = 57.1426{{c}}, ~131072/91875 = 614.9336{{c}}
: [[error map]]: {{val| +0.030 -0.074 -0.010 +0.059 }}
: [[error map]]: {{val| -0.005 -0.012 +0.039 -0.013 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~243/224 = 140.3685{{c}}
* [[CWE]]: ~1323/1280 = 57.1429{{c}}, ~131072/91875 = 614.9360{{c}}
: error map: {{val| 0.000 -0.113 -0.050 +0.017 }}
: error map: {{val| 0.000 -0.004 +0.051 +0.002 }}
 
{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}


{{Optimal ET sequence|legend=1| 60, 111, 171, 795, 966, 1137, 1308, 5403b, 6711b, 8019bc }}
[[Badness]] (Sintel): 0.838


[[Badness]] (Sintel): 0.426
=== Vasca ===
Vasca can be described as the {{nowrap| 357 & 525 }} temperament, extended as high as the 23-limit. It tempers out the {{monzo| 95 0 0 0 0 0 0 0 -21 }}, and sets a stack of twenty-one [[23/16]]'s equal with eleven octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.  


=== Semiseptichrome ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 9801/9800, 151263/151250, 234375/234256
Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}


Mapping: {{mapping| 6 1 -15 -24 -32 | 0 5 17 24 31 }}
{{Mapping|legend=0| 21 4 39 98 58 | 0 6 2 -8 3 3 }}
: mapping generators: ~55/49, ~375/308
: mapping generators: ~1323/1280, ~6615/5632


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 200.0058{{c}}, ~375/308 = 340.3742{{c}} (~1760/1701 = 59.6375{{c}})
* WE: ~1323/1280 = 57.1436{{c}}, ~6615/5632 = 278.9017{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3661{{c}} (~1760/1701 = 59.6339{{c}})
* CWE: ~1323/1280 = 57.1429{{c}}, ~6615/5632 = 278.8985{{c}}


{{Optimal ET sequence|legend=0| 60e, 222cdee, 282, 342, 966, 1308, 1650, 4608b, 6258bc }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882, 1407, 2289e }}


Badness (Sintel): 0.642
Badness (Sintel): 3.14


==== 13-limit ====
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1716/1715, 2080/2079, 34398/34375, 85293/85184
Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683


Mapping: {{mapping| 6 1 -15 -24 -32 -68 | 0 5 17 24 31 53 }}
{{Mapping|legend=0| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9936{{c}}, ~375/308 = 340.3707{{c}} (~121/117 = 59.6165{{c}})
* WE: ~336/325 = 57.1426{{c}}, ~168/143 = 278.9047{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3802{{c}} (~121/117 = 59.6198{{c}})
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9060{{c}}


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}


Badness (Sintel): 1.64
Badness (Sintel): 2.28


==== 17-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 61965/61952
Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625


Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 | 0 5 17 24 31 53 15 }}
{{Mapping|legend=0| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9865{{c}}, ~375/308 = 340.3619{{c}} (~88/85 = 59.6111{{c}})
* WE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9037{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~375/308 = 340.3821{{c}} (~88/85 = 59.6179{{c}})
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}


Badness (Sintel): 1.39
Badness (Sintel): 1.62


==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 23409/23408
Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095


Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 | 0 5 17 24 31 53 15 -5 }}
{{Mapping|legend=0| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9837{{c}}, ~162/133 = 340.3589{{c}} (~88/85 = 59.6084{{c}})
* WE: ~336/325 = 57.1425{{c}}, ~168/143 = 278.8960{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8976{{c}}


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}


Badness (Sintel): 1.35
Badness (Sintel): 1.64


==== 23-limit ====
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 936/935, 1701/1700, 1716/1715, 1863/1862, 2024/2023, 2025/2023, 2376/2375
Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211


Mapping: {{mapping| 6 1 -15 -24 -32 -68 -1 34 -12 | 0 5 17 24 31 53 15 -5 23 }}
{{Mapping|legend=0| 21 4 39 98 58 107 120 16 95 | 0 6 2 -8 3 -6 -7 15 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~55/49 = 199.9829{{c}}, ~162/133 = 340.3576{{c}} (~88/85 = 59.6081{{c}})
* WE: ~336/325 = 57.1422{{c}}, ~168/143 = 278.8949{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~162/133 = 340.3844{{c}} (~88/85 = 59.6156{{c}})
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8980{{c}}


{{Optimal ET sequence|legend=0| 282, 342f, 624 }}
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}


Badness (Sintel): 1.14
Badness (Sintel): 1.43


== Pnict ==
== Pnict ==
Line 265: Line 242:


Atomic tempers out the [[atom]], {{monzo| 161 -84 -12 }}, and in the [[7-limit]] the [[nommisma]], {{monzo| -55 30 2 1 }}, so that a stack of two [[schisma]]s gives the [[garischisma]], from which intervals of [[7/1|7]] can be derived. It may be described as the {{nowrap| 12 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid monocot.  
Atomic tempers out the [[atom]], {{monzo| 161 -84 -12 }}, and in the [[7-limit]] the [[nommisma]], {{monzo| -55 30 2 1 }}, so that a stack of two [[schisma]]s gives the [[garischisma]], from which intervals of [[7/1|7]] can be derived. It may be described as the {{nowrap| 12 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid monocot.  
Atomic extensions discussed elsewhere include [[72nd-octave temperaments #Hafnium|hafnium]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 289: Line 264:
Comma list: 9801/9800, 151263/151250, 184549376/184528125
Comma list: 9801/9800, 151263/151250, 184549376/184528125


Mapping: {{mapping| 12 0 161 338 517 | 0 1 -7 -16 -25 }}
{{Mapping|legend=0| 12 0 161 338 517 | 0 1 -7 -16 -25 }}


Optimal tunings:  
Optimal tunings:  
Line 306: Line 281:
Comma list: 9801/9800, 151263/151250, 371293/371250, 184549376/184528125
Comma list: 9801/9800, 151263/151250, 371293/371250, 184549376/184528125


Mapping: {{mapping| 60 0 805 1690 2585 1173 | 0 1 7 -16 -25 -10 }}
{{Mapping|legend=0| 60 0 805 1690 2585 1173 | 0 1 7 -16 -25 -10 }}
: mapping generators: ~2704/2673, ~3
: mapping generators: ~2704/2673, ~3


Line 322: Line 297:
Comma list: 9801/9800, 12376/12375, 28561/28560, 151263/151250, 11275335/11275264
Comma list: 9801/9800, 12376/12375, 28561/28560, 151263/151250, 11275335/11275264


Mapping: {{mapping| 60 0 805 1690 2585 1173 1957 | 0 1 7 -16 -25 -10 -18 }}
{{Mapping|legend=0| 60 0 805 1690 2585 1173 1957 | 0 1 7 -16 -25 -10 -18 }}


Optimal tunings:  
Optimal tunings:  
Line 337: Line 312:
Comma list: 9801/9800, 12376/12375, 12636/12635, 23409/23408, 28561/28560, 151263/151250
Comma list: 9801/9800, 12376/12375, 12636/12635, 23409/23408, 28561/28560, 151263/151250


Mapping: {{mapping| 60 60 385 730 1085 573 877 -61 | 0 1 7 -16 -25 -10 -18 9 }}
{{Mapping|legend=0| 60 60 385 730 1085 573 877 -61 | 0 1 7 -16 -25 -10 -18 9 }}


Optimal tunings:  
Optimal tunings:  
Line 348: Line 323:


==== Hafnium ====
==== Hafnium ====
Hafnium (1224 & 4320), named after the 72nd element, splits the 1/12-octave period into six 1/72-octave parts. Since 4320edo and 5544edo have good 31st and 37th harmonics, addition of these primes are also prescribed. In the add-37 version, [[37/22]] is mapped to exact 3\4.  
Hafnium (1224 & 4320), named after the 72nd element, splits the 1/12-octave period into six 1/72-octave parts. Since 4320edo and 5544edo have good 31st and 37th harmonics, addition of these primes are also prescribed. In the add-37 version, [[37/22]] is mapped to exact 3\4 (see also [[Berylic]]).


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
Line 354: Line 329:
Comma list: 9801/9800, 151263/151250, 184549376/184528125, 308915776/308828625
Comma list: 9801/9800, 151263/151250, 184549376/184528125, 308915776/308828625


Mapping: {{mapping| 72 0 966 2028 3102 2777 | 0 1 -7 -16 -25 -22 }}
{{Mapping|legend=0| 72 0 966 2028 3102 2777 | 0 1 -7 -16 -25 -22 }}
: mapping generators: ~105/104, ~3
: mapping generators: ~105/104, ~3


Line 370: Line 345:
Comma list: 9801/9800, 12376/12375, 151263/151250, 1713660/1713481, 97144749/97140736
Comma list: 9801/9800, 12376/12375, 151263/151250, 1713660/1713481, 97144749/97140736


Mapping: {{mapping| 72 0 966 2028 3102 2777 979 | 0 1 -7 -16 -25 -22 -6 }}
{{Mapping|legend=0| 72 0 966 2028 3102 2777 979 | 0 1 -7 -16 -25 -22 -6 }}
: mapping generators: ~105/104, ~3
: mapping generators: ~105/104, ~3


Line 386: Line 361:
Comma list: 9801/9800, 10881/10880, 12376/12375, 57629/57624, 179712/179707, 61456384/61448625
Comma list: 9801/9800, 10881/10880, 12376/12375, 57629/57624, 179712/179707, 61456384/61448625


Subgroup-val mapping: {{mapping| 72 0 966 2028 3102 2777 979 -328 | 0 1 -7 -16 -25 -22 -6 6 }}
{{Mapping|legend=2| 72 0 966 2028 3102 2777 979 -328 | 0 1 -7 -16 -25 -22 -6 6 }}


Optimal tunings:  
Optimal tunings:  
Line 401: Line 376:
Comma list: 9801/9800, 10881/10880, 12376/12375, 16576/16575, 57629/57624, 93093/93092, 179712/179707
Comma list: 9801/9800, 10881/10880, 12376/12375, 16576/16575, 57629/57624, 93093/93092, 179712/179707


Subgroup-val mapping: {{mapping| 72 0 966 2028 3102 2777 979 -328 3228 | 0 1 -7 -16 -25 -22 -6 6 -25 }}
{{Mapping|legend=2| 72 0 966 2028 3102 2777 979 -328 3228 | 0 1 -7 -16 -25 -22 -6 6 -25 }}


Optimal tunings:  
Optimal tunings:  
Line 410: Line 385:


Badness (Sintel): 1.98
Badness (Sintel): 1.98
== Slendscape ==
Named by [[Xenllium]] in 2025, slendscape tempers out the [[slendroschisma]] (68719476736/68641485507) in addition to landscape comma, and thereby features a period of 1\15.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 250047/250000, 12884901888/12867859375
{{Mapping|legend=1| 15 0 17 54 | 0 4 3 -2 }}
: mapping generators: ~8575/8192, ~1152/875
[[Optimal tuning]]s:
* [[WE]]: ~8575/8192 = 79.9771{{c}}, ~1152/875 = 475.4832{{c}}
: [[error map]]: {{val| -0.043 -0.022 +0.087 +0.053 }}
* [[CWE]]: ~8575/8192 = 80.0000{{c}}, ~1152/875 = 475.4962{{c}}
: error map: {{val| 0.000 +0.030 +0.175 +0.182 }}
{{Optimal ET sequence|legend=1| 15, 240, 255, 270, 795, 1065, 1335, 2400 }}
[[Badness]] (Sintel): 1.47
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 102487/102400, 180224/180075
{{Mapping|legend=0| 15 0 17 54 40 | 0 4 3 -2 2 }}
Optimal tunings:
* WE: ~22/21 = 79.9991{{c}}, ~968/735 = 475.4915{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~968/735 = 475.4955{{c}}
{{Optimal ET sequence|legend=0| 15, 240, 255, 270, 795, 1065, 2400e }}
Badness (Sintel): 0.868
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625
{{Mapping|legend=0| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}
Optimal tunings:
* WE: ~22/21 = 79.9993{{c}}, ~154/117 = 475.4902{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~154/117 = 475.4943{{c}}
{{Optimal ET sequence|legend=0| 255, 270, 795, 1065 }}
Badness (Sintel): 0.877


== Avicenna ==
== Avicenna ==
Line 434: Line 459:
Comma list: 3025/3024, 5632/5625, 102487/102400
Comma list: 3025/3024, 5632/5625, 102487/102400


Mapping: {{mapping| 3 2 8 16 9 | 0 8 -3 -22 4 }}
{{Mapping|legend=0| 3 2 8 16 9 | 0 8 -3 -22 4 }}


Optimal tunings:  
Optimal tunings:  
Line 449: Line 474:
Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095
Comma list: 676/675, 1001/1000, 3025/3024, 4096/4095


Mapping: {{mapping| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}
{{Mapping|legend=0| 3 2 8 16 9 8 | 0 8 -3 -22 4 9 }}


Optimal tunings:  
Optimal tunings:  
Line 464: Line 489:
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4096/4095
Comma list: 676/675, 715/714, 936/935, 1001/1000, 4096/4095


Mapping: {{mapping| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}
{{Mapping|legend=0| 3 2 8 16 9 8 4 | 0 8 -3 -22 4 9 24 }}


Optimal tunings:  
Optimal tunings:  
Line 479: Line 504:
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1729/1728
Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1729/1728


Mapping: {{mapping| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}
{{Mapping|legend=0| 3 2 8 16 9 8 4 0 | 0 8 -3 -22 4 9 24 37 }}


Optimal tunings:  
Optimal tunings:  
Line 489: Line 514:
Badness (Sintel): 0.928
Badness (Sintel): 0.928


== Terture ==
== Zinc ==
{{See also| Vulture family }}
Zinc maybe described as the {{nowrap| 270 & 2190 }} temperament, and is a [[weak extension]] of [[miscellaneous 5-limit temperaments #Neon|neon]]. It was named by [[Eliora]] in 2023 after the 30th element for having a 30th-octave period.
 
Named by [[Xenllium]] in 2021, terture may be described as the {{nowrap| 111 & 159 }} temperament.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 359661568/358722675
[[Comma list]]: 250047/250000, {{monzo| -53 -12 2 24 }}


{{Mapping|legend=1| 3 0 -18 -32 | 0 4 21 34 }}
{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}
: mapping generators: ~63/50, ~320/243
: mapping generators: ~53747712/52521875, ~216/175


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 399.9723{{c}}, ~320/243 = 475.5221{{c}} (~392/375 = 75.5499{{c}})
* [[WE]]: ~53747712/52521875 = 40.0002{{c}}, ~216/175 = 364.3890{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3869{{c}})
: [[error map]]: {{val| -0.083 +0.134 +0.151 -0.185 }}
: [[error map]]: {{val| +0.007 -0.009 +0.024 -0.032 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5519{{c}} (~392/375 = 75.5519{{c}})
* [[CWE]]: ~53747712/52521875 = 40.0000{{c}}, ~216/175 = 364.3879{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3879{{c}})
: error map: {{val| 0.000 +0.253 +0.276 -0.061 }}
: error map: {{val| 0.000 -0.015 +0.014 -0.050 }}


{{Optimal ET sequence|legend=1| 111, 159, 270 }}
{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}


[[Badness]] (Sintel): 2.21
[[Badness]] (Sintel): 1.88


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 19712/19683, 102487/102400
Comma list: 9801/9800, 151263/151250, {{monzo| -27 -6 4 6 3 }}


Mapping: {{mapping| 3 0 -18 -32 8 | 0 4 21 34 2 }}
{{Mapping|legend=0| 30 2 15 66 122 | 0 5 6 2 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9902{{c}}, ~320/243 = 475.5383{{c}} (~392/375 = 75.5481{{c}})
* WE: ~18865/18432 = 40.0005{{c}}, ~216/175 = 364.3881{{c}} (~385/384 = 4.3837{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~320/243 = 475.5490{{c}} (~392/375 = 75.5490{{c}})
* CWE: ~18865/18432 = 40.0000{{c}}, ~216/175 = 364.3849{{c}} (~385/384 = 4.3849{{c}})


{{Optimal ET sequence|legend=0| 111, 159, 270, 1239, 1509, 1779, 2049, 2319 }}
{{Optimal ET sequence|legend=0| 270, 1110, 1380, 1650, 1920, 2190 }}


Badness (Sintel): 0.969
Badness (Sintel): 0.727


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 676/675, 1001/1000, 3025/3024, 10985/10976
Comma list: 9801/9800, 10648/10647, 105644/105625, 196625/196608


Mapping: {{mapping| 3 0 -18 -32 8 -21 | 0 4 21 34 2 27 }}
{{Mapping|legend=0| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~63/50 = 399.9958{{c}}, ~154/117 = 475.5485{{c}} (~117/112 = 75.5527{{c}})
* WE: ~351/343 = 40.0003{{c}}, ~216/175 = 364.3894{{c}} (~385/384 = 4.3865{{c}})
* CWE: ~63/50 = 400.0000{{c}}, ~154/117 = 475.5531{{c}} (~117/112 = 75.5531{{c}})
* CWE: ~351/343 = 40.0000{{c}}, ~216/175 = 364.3867{{c}} (~385/384 = 4.3867{{c}})


{{Optimal ET sequence|legend=0| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}
 
Badness (Sintel): 0.640


Badness (Sintel): 0.771
==== 2.3.5.7.11.13.19 subgroup ====
A 2.7.11.19 restriction worth considering is [[30th-octave temperaments#Oyster|oyster]].


=== 17-limit ===
Subgroup: 2.3.5.7.11.13.19
Subgroup: 2.3.5.7.11.13.17


Comma list: 676/675, 715/714, 936/935, 1001/1000, 4928/4913
Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 89376/89375


Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 | 0 4 21 34 2 27 12 }}
{{Mapping|legend=0| 30 2 15 66 122 193 91 | 0 5 6 2 -2 -9 4 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~34/27 = 399.9664{{c}}, ~112/85 = 475.5198{{c}} (~117/112 = 75.5534{{c}})
* WE: ~175/171 = 40.0002{{c}}, ~216/175 = 364.3885{{c}} (~400/399 = 4.3862{{c}})
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~117/112 = 75.5568{{c}})
* CWE: ~175/171 = 40.0000{{c}}, ~216/175 = 364.3864{{c}} (~400/399 = 4.3864{{c}})


{{Optimal ET sequence|legend=0| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e }}


Badness (Sintel): 0.953
Badness (Sintel): 0.477


=== 19-limit ===
=== Neodymium ===
Subgroup: 2.3.5.7.11.13.17.19
Neodymium (540 & 1920) splits the period into 1/60-octave halves for prime 17.
 
Subgroup: 2.3.5.7.11.13.17


Comma list: 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1617/1615
Comma list: 9801/9800, 10648/10647, 31213/31212, 105644/105625, 196625/196608


Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 | 0 4 21 34 2 27 12 41 }}
{{Mapping|legend=0| 60 4 30 132 244 386 391 | 0 5 6 2 -2 -9 -8 }}
: mapping generators: ~612/605, ~216/175


Optimal tunings:  
Optimal tunings:  
* WE: ~34/27 = 399.9665{{c}}, ~112/85 = 475.5198{{c}} (~95/91 = 75.5533{{c}})
* WE: ~612/605 = 20.0002{{c}}, ~216/175 = 364.3894{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5568{{c}} (~95/91 = 75.5568{{c}})
* CWE: ~612/605 = 20.0000{{c}}, ~216/175 = 364.3864{{c}}


{{Optimal ET sequence|legend=0| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 540, 1380, 1920, 2460, 4380, 6840e }}


Badness (Sintel): 0.846
Badness (Sintel): 1.23


=== 23-limit ===
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 460/459, 529/528, 676/675, 715/714, 936/935, 1001/1000, 1216/1215
Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 23409/23408, 89376/89375


Mapping: {{mapping| 3 0 -18 -32 8 -21 -2 -36 10 | 0 4 21 34 2 27 12 41 3 }}
{{Mapping|legend=0| 60 4 30 132 244 386 391 182 | 0 5 6 2 -2 -9 -8 4 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~34/27 = 400.0026{{c}}, ~112/85 = 475.5510{{c}} (~24/23 = 75.5485{{c}})
* WE: ~612/605 = 20.0001{{c}}, ~216/175 = 364.3883{{c}}
* CWE: ~34/27 = 400.0000{{c}}, ~112/85 = 475.5482{{c}} (~24/23 = 75.5482{{c}})
* CWE: ~612/605 = 20.0000{{c}}, ~216/175 = 364.3860{{c}}


{{Optimal ET sequence|legend=0| 111, 159, 270 }}
{{Optimal ET sequence|legend=0| 540, 1380, 1920, 2460, 4380, 6840e }}


Badness (Sintel): 1.07
Badness (Sintel): 1.02
 
== Magnesium ==
: ''For the 5-limit version, see [[12th-octave temperaments #Magnesium (5-limit)]].


== Slendscape ==
Magnesium is named by [[Eliora]] in 2023 after the 12th element for having a 1/12-octave period; however, it is not an extension of the [[atomic]] – the associated comma is {{monzo| -157 -24 84 }} in the 5-limit, and 7 generator steps together with two [[12edo]] semitones reach the [[3/1|3rd]] [[harmonic]]. It may be described as the {{nowrap| 84 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid gamma-heptacot.  
Named by [[Xenllium]] in 2025, slendscape tempers out the [[slendroschisma]] (68719476736/68641485507) in addition to landscape comma, and thereby features a period of 1\15.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 12884901888/12867859375
[[Comma list]]: 250047/250000, {{monzo| -59 2 18 5 }}


{{Mapping|legend=1| 15 0 17 54 | 0 4 3 -2 }}
{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
: mapping generators: ~8575/8192, ~1152/875
: mapping generators: ~138915/131072, ~3145728/2734375


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~8575/8192 = 79.9771{{c}}, ~1152/875 = 475.4832{{c}}
* [[WE]]: ~138915/131072 = 100.0021{{c}}, ~3145728/2734375 = 243.1333{{c}}
: [[error map]]: {{val| -0.043 -0.022 +0.087 +0.053 }}
: [[error map]]: {{val| +0.025 -0.018 +0.000 -0.039 }}
* [[CWE]]: ~8575/8192 = 80.0000{{c}}, ~1152/875 = 475.4962{{c}}
* [[CWE]]: ~138915/131072 = 100.0000{{c}}, ~3145728/2734375 = 243.1285{{c}}
: error map: {{val| 0.000 +0.030 +0.175 +0.182 }}
: error map: {{val| 0.000 -0.055 -0.057 -0.111 }}


{{Optimal ET sequence|legend=1| 15, 240, 255, 270, 795, 1065, 1335, 2400 }}
{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}


[[Badness]] (Sintel): 1.47
Badness (Sintel): 2.44


=== 11-limit ===
== Poe ==
Subgroup: 2.3.5.7.11
Named by [[Tristan Bay]] in 2025, poe may be described as the {{nowrap| 60 & 270 }} temperament.  
 
Comma list: 3025/3024, 102487/102400, 180224/180075
 
Mapping: {{mapping| 15 0 17 54 40 | 0 4 3 -2 2 }}
 
Optimal tunings:
* WE: ~22/21 = 79.9991{{c}}, ~968/735 = 475.4915{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~968/735 = 475.4955{{c}}
 
{{Optimal ET sequence|legend=0| 15, 240, 255, 270, 795, 1065, 2400e }}
 
Badness (Sintel): 0.868
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1716/1715, 3025/3024, 4096/4095, 14641/14625
 
Mapping: {{mapping| 15 0 17 54 40 109 | 0 4 3 -2 2 -9 }}
 
Optimal tunings:
* WE: ~22/21 = 79.9993{{c}}, ~154/117 = 475.4902{{c}}
* CWE: ~22/21 = 80.0000{{c}}, ~154/117 = 475.4943{{c}}
 
{{Optimal ET sequence|legend=0| 255, 270, 795, 1065 }}
 
Badness (Sintel): 0.877
 
== Akjayland ==
{{See also| 21st-octave temperaments }}
 
Named by [[Eliora]] in 2022, akjayland tempers out the [[akjaysma]] in addition to landscape comma, and thereby features a period of 1\21.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, {{monzo| 43 -1 -13 -4 }}
[[Comma list]]: 250047/250000, {{monzo| 15 -16 -4 7 }}


{{Mapping|legend=1| 21 1 38 102 | 0 3 1 -4 }}
{{Mapping|legend=1| 30 0 -73 -106 | 0 1 3 4 }}
: mapping generators: ~1323/1280, ~131072/91875
: mapping generators: ~2240/2187, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~1323/1280 = 57.1426{{c}}, ~131072/91875 = 614.9336{{c}}
* [[WE]]: ~2240/2187 = 39.9982{{c}}, ~3/2 = 702.1533{{c}}
: [[error map]]: {{val| -0.005 -0.012 +0.039 -0.013 }}
: [[error map]]: {{val| -0.055 +0.143 +0.115 -0.238 }}
* [[CWE]]: ~1323/1280 = 57.1429{{c}}, ~131072/91875 = 614.9360{{c}}
* [[CWE]]: ~2240/2187 = 40.0000{{c}}, ~3/2 = 702.1656{{c}}
: error map: {{val| 0.000 -0.004 +0.051 +0.002 }}
: error map: {{val| 0.000 +0.211 +0.183 -0.163 }}


{{Optimal ET sequence|legend=1| 84, 273, 357, 441, 966, 1407, 1848, 7833, 9681, 11529, 13377c }}
{{Optimal ET sequence|legend=1| 60, 150cd, 210, 270, 1950, 2220, 2490, 2760b, 3030bc, 3300bc }}


[[Badness]] (Sintel): 0.838
[[Badness]] (Sintel): 2.90
 
=== Vasca ===
Vasca can be described as the {{nowrap| 357 & 525 }} temperament, extended as high as the 23-limit. It tempers out the {{monzo| 95 0 0 0 0 0 0 0 -21 }}, and sets a stack of twenty-one [[23/16]]'s equal with eleven octaves. The name derives from elements vanadium (23) and scandium (21), since this uses the 23rd harmonic, which itself is extremely well represented in 21edo.  


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 3025/3024, 102487/102400, {{monzo| 39 -4 -11 -5 2 }}
Comma list: 9801/9800, 19712/19683, 151263/151250


Mapping: {{mapping| 21 4 39 98 58 | 0 6 2 -8 3 3 }}
{{Mapping|legend=0| 30 0 -73 -106 -134 | 0 1 3 4 5 }}
: mapping generators: ~1323/1280, ~6615/5632


Optimal tunings:  
Optimal tunings:  
* WE: ~1323/1280 = 57.1436{{c}}, ~6615/5632 = 278.9017{{c}}
* WE: ~45/44 = 39.9976{{c}}, ~3/2 = 702.1955{{c}}
* CWE: ~1323/1280 = 57.1429{{c}}, ~6615/5632 = 278.8985{{c}}
* CWE: ~45/44 = 40.0000{{c}}, ~3/2 = 702.2129{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882, 1407, 2289e }}
{{Optimal ET sequence|legend=0| 60e, …, 210e, 270 }}


Badness (Sintel): 3.14
Badness (Sintel): 1.31


==== 13-limit ====
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 3025/3024, 4096/4095, 14641/14625, 85750/85683
Comma list: 1001/1000, 4225/4224, 4459/4455, 19712/19683


Mapping: {{mapping| 21 4 39 98 58 107 | 0 6 2 -8 3 -6 }}
{{Mapping|legend=0| 30 0 -73 -106 -134 111 | 0 1 3 4 5 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~336/325 = 57.1426{{c}}, ~168/143 = 278.9047{{c}}
* WE: ~45/44 = 40.0008{{c}}, ~3/2 = 702.1778{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9060{{c}}
* CWE: ~45/44 = 40.0000{{c}}, ~3/2 = 702.1699{{c}}


{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}
{{Optimal ET sequence|legend=0| 60e, 210e, 270, 1410ef, 1680ef }}


Badness (Sintel): 2.28
Badness (Sintel): 1.19
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 2601/2600, 3025/3024, 4096/4095, 8624/8619, 14641/14625
 
Mapping: {{mapping| 21 4 39 98 58 107 120 | 0 6 2 -8 3 -6 -7 }}
 
Optimal tunings:
* WE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9037{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.9036{{c}}
 
{{Optimal ET sequence|legend=0| 168, 357, 525, 882 }}
 
Badness (Sintel): 1.62
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 2376/2375, 2601/2600, 2926/2925, 3025/3024, 3213/3211, 4096/4095
 
Mapping: {{mapping| 21 4 39 98 58 107 120 16 | 0 6 2 -8 3 -6 -7 15 }}
 
Optimal tunings:
* WE: ~336/325 = 57.1425{{c}}, ~168/143 = 278.8960{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8976{{c}}
 
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}
 
Badness (Sintel): 1.64
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 1496/1495, 2376/2375, 2601/2600, 2646/2645, 2926/2925, 3025/3024, 3213/3211
 
Mapping: {{mapping| 21 4 39 98 58 107 120 16 95 | 0 6 2 -8 3 -6 -7 15 0 }}
 
Optimal tunings:
* WE: ~336/325 = 57.1422{{c}}, ~168/143 = 278.8949{{c}}
* CWE: ~336/325 = 57.1429{{c}}, ~168/143 = 278.8980{{c}}
 
{{Optimal ET sequence|legend=0| 168h, 357, 525, 882, 1407 }}
 
Badness (Sintel): 1.43
 
== Magnesium ==
: ''For the 5-limit version, see [[12th-octave temperaments #Magnesium (5-limit)]].
 
Magnesium is named by [[Eliora]] in 2023 after the 12th element for having a 1/12-octave period; however, it is not an extension of the [[atomic]] – the associated comma is {{monzo| -157 -24 84 }} in the 5-limit, and 7 generator steps together with two [[12edo]] semitones reach the [[3/1|3rd]] [[harmonic]]. It may be described as the {{nowrap| 84 & 612 }} temperament, with a [[ploidacot]] signature of dodecaploid gamma-heptacot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, {{monzo| -59 2 18 5 }}
 
{{Mapping|legend=1| 12 2 23 58 | 0 7 2 -10 }}
: mapping generators: ~138915/131072, ~3145728/2734375
 
[[Optimal tuning]]s:
* [[WE]]: ~138915/131072 = 100.0021{{c}}, ~3145728/2734375 = 243.1333{{c}}
: [[error map]]: {{val| +0.025 -0.018 +0.000 -0.039 }}
* [[CWE]]: ~138915/131072 = 100.0000{{c}}, ~3145728/2734375 = 243.1285{{c}}
: error map: {{val| 0.000 -0.055 -0.057 -0.111 }}
 
{{Optimal ET sequence|legend=1| 84, 360d, 444, 528, 612, 1920, 2532, 10740cd, 13272bcdd, 15804bcdd, 18336bcddd }}
 
Badness (Sintel): 2.44


== Chromium ==
== Chromium ==
Line 785: Line 713:
Comma list: 9801/9800, 46656/46585, 151263/151250
Comma list: 9801/9800, 46656/46585, 151263/151250


Mapping: {{mapping| 24 1 -6 18 46 | 0 3 5 4 3 }}
{{Mapping|legend=0| 24 1 -6 18 46 | 0 3 5 4 3 }}


Optimal tunings:  
Optimal tunings:  
Line 800: Line 728:
Comma list: 1716/1715, 2080/2079, 34398/34375, 39366/39325
Comma list: 1716/1715, 2080/2079, 34398/34375, 39366/39325


Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 }}
{{Mapping|legend=0| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 }}


Optimal tunings:  
Optimal tunings:  
Line 815: Line 743:
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 11016/11011
Comma list: 936/935, 1701/1700, 1716/1715, 2025/2023, 11016/11011


Mapping: {{mapping| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 9 }}
{{Mapping|legend=0| 24 1 -6 18 46 -47 -13 | 0 3 5 4 3 11 9 }}


Optimal tunings:  
Optimal tunings:  
Line 825: Line 753:
Badness (Sintel): 1.06
Badness (Sintel): 1.06


== Zinc ==
== Phosphorus ==
{{See also| 10th-octave temperaments #Neon | 60th-octave temperaments #Neodymium }}
 
Zinc maybe described as the {{nowrap| 270 & 2190 }} temperament. It was named by [[Eliora]] in 2023 after the 30th element for having a 30th-octave period.
 
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, {{monzo| -53 -12 2 24 }}
[[Comma list]]: 250047/250000, {{monzo| 60 -96 1 32 }}


{{Mapping|legend=1| 30 2 15 66 | 0 5 6 2 }}
[[Mapping]]: {{mapping| 15 8 -4 -4 | 0 13 32 38 }}  
: mapping generators: ~53747712/52521875, ~216/175
: mapping generators: ~{{monzo| -21 34 4 -15 }}, ~210827008/199290375


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~53747712/52521875 = 40.0002{{c}}, ~216/175 = 364.3890{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3869{{c}})
* [[WE]]: ~{{monzo|-21 34 4 -15}} = 80.000039{{c}}, ~210827008/199290375 = 97.073339{{c}}
: [[error map]]: {{val| +0.007 -0.009 +0.024 -0.032 }}
: [[error map]]: {{val| +0.0006 -0.0013 +0.0330 -0.0392 }}
* [[CWE]]: ~53747712/52521875 = 40.0000{{c}}, ~216/175 = 364.3879{{c}} (~{{monzo| 21 3 1 -10 }} = 4.3879{{c}})
* [[CWE]]: ~{{monzo|-21 34 4 -15}} = 80.000000{{c}}, ~210827008/199290375 = 97.073314{{c}}
: error map: {{val| 0.000 -0.015 +0.014 -0.050 }}
: error map: {{val| 0.0000 -0.0019 +0.0323 -0.0400 }}


{{Optimal ET sequence|legend=1| 270, 1380, 1650, 1920, 2190, 4650 }}
{{Optimal ET sequence|legend=1| 210, …, 1125, 1335, 2460 }}


[[Badness]] (Sintel): 1.88
[[Badness]] (Sintel): 9.66
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 151263/151250, {{monzo| -27 -6 4 6 3 }}
 
Mapping: {{mapping| 30 2 15 66 122 | 0 5 6 2 -2 }}
 
Optimal tunings:
* WE: ~18865/18432 = 40.0005{{c}}, ~216/175 = 364.3881{{c}} (~385/384 = 4.3837{{c}})
* CWE: ~18865/18432 = 40.0000{{c}}, ~216/175 = 364.3849{{c}} (~385/384 = 4.3849{{c}})
 
{{Optimal ET sequence|legend=0| 270, 1110, 1380, 1650, 1920, 2190 }}
 
Badness (Sintel): 0.727
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 9801/9800, 10648/10647, 105644/105625, 196625/196608
 
Mapping: {{mapping| 30 2 15 66 122 193 | 0 5 6 2 -2 -9 }}
 
Optimal tunings:
* WE: ~351/343 = 40.0003{{c}}, ~216/175 = 364.3894{{c}} (~385/384 = 4.3865{{c}})
* CWE: ~351/343 = 40.0000{{c}}, ~216/175 = 364.3867{{c}} (~385/384 = 4.3867{{c}})
 
{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e, 11490de }}
 
Badness (Sintel): 0.640
 
==== 2.3.5.7.11.13.19 subgroup (neozinc) ====
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 89376/89375
 
Mapping: {{mapping| 30 2 15 66 122 193 91 | 0 5 6 2 -2 -9 4 }}
 
Optimal tunings:
* WE: ~175/171 = 40.0002{{c}}, ~216/175 = 364.3885{{c}} (~400/399 = 4.3862{{c}})
* CWE: ~175/171 = 40.0000{{c}}, ~216/175 = 364.3864{{c}} (~400/399 = 4.3864{{c}})
 
{{Optimal ET sequence|legend=0| 270, 1380, 1650, 1920, 2190, 4650, 6840e }}
 
Badness (Sintel): 0.477
 
=== Neodymium ===
Neodymium (540 & 1920) splits the period into 1/60-octave halves for prime 17.
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 9801/9800, 10648/10647, 31213/31212, 105644/105625, 196625/196608
 
Mapping: {{mapping| 60 4 30 132 244 386 391 | 0 5 6 2 -2 -9 -8 }}
: mapping generators: ~612/605, ~216/175
 
Optimal tunings:
* WE: ~612/605 = 20.0002{{c}}, ~216/175 = 364.3894{{c}}
* CWE: ~612/605 = 20.0000{{c}}, ~216/175 = 364.3864{{c}}
 
{{Optimal ET sequence|legend=0| 540, 1380, 1920, 2460, 4380, 6840e }}
 
Badness (Sintel): 1.23
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 5929/5928, 6860/6859, 9801/9800, 10241/10240, 23409/23408, 89376/89375
 
Mapping: {{mapping| 60 4 30 132 244 386 391 182 | 0 5 6 2 -2 -9 -8 4 }}
 
Optimal tunings:
* WE: ~612/605 = 20.0001{{c}}, ~216/175 = 364.3883{{c}}
* CWE: ~612/605 = 20.0000{{c}}, ~216/175 = 364.3860{{c}}
 
{{Optimal ET sequence|legend=0| 540, 1380, 1920, 2460, 4380, 6840e }}
 
Badness (Sintel): 1.02
 
== Poe ==
Named by [[Tristan Bay]] in 2025, poe may be described as the {{nowrap| 60 & 270 }} temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 250047/250000, {{monzo| 15 -16 -4 7 }}
 
{{Mapping|legend=1| 30 0 -73 -106 | 0 1 3 4 }}
: mapping generators: ~2240/2187, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2240/2187 = 39.9982{{c}}, ~3/2 = 702.1533{{c}}
: [[error map]]: {{val| -0.055 +0.143 +0.115 -0.238 }}
* [[CWE]]: ~2240/2187 = 40.0000{{c}}, ~3/2 = 702.1656{{c}}
: error map: {{val| 0.000 +0.211 +0.183 -0.163 }}
 
{{Optimal ET sequence|legend=1| 60, 150cd, 210, 270, 1950, 2220, 2490, 2760b, 3030bc, 3300bc }}
 
[[Badness]] (Sintel): 2.90
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 19712/19683, 151263/151250
 
Mapping: {{mapping| 30 0 -73 -106 -134 | 0 1 3 4 5 }}
 
Optimal tunings:
* WE: ~45/44 = 39.9976{{c}}, ~3/2 = 702.1955{{c}}
* CWE: ~45/44 = 40.0000{{c}}, ~3/2 = 702.2129{{c}}
 
{{Optimal ET sequence|legend=0| 60e, …, 210e, 270 }}
 
Badness (Sintel): 1.31
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 4225/4224, 4459/4455, 19712/19683
 
Mapping: {{mapping| 30 0 -73 -106 -134 111 | 0 1 3 4 5 0 }}
 
Optimal tunings:
* WE: ~45/44 = 40.0008{{c}}, ~3/2 = 702.1778{{c}}
* CWE: ~45/44 = 40.0000{{c}}, ~3/2 = 702.1699{{c}}
 
{{Optimal ET sequence|legend=0| 60e, 210e, 270, 1410ef, 1680ef }}
 
Badness (Sintel): 1.19


[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Landscape| ]] <!-- key article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Landscape microtemperaments| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Landscape_microtemperaments"